"On this matter" is correct.
"On this matter" means "about this subject" and is very fitting here.
"In this regard" is used as reference to something previously mentioned. It would refer back to something said before the sentence in your question. Do you have the sentence that came before this one?
"master's" is correct. "master's" is short for "master's degree". The degree belongs to the master so you use the possessive 's here. "She graduated with a master's (degree) (in Chemistry)
If you are referring to a specific degree then you don't use the 's and it must be capitalised. For example, "She graduated with a Master of Sciences" (Master of Sciences is usually abbreviated to MSc).